Multiobjective Metaheuristics Optimization in Reactive Power Compensation

Transcription

1 Ilana González Palau, Secundno Marrero Ramírez, Arístdes Legrá Lobana, Danel Mendola Ells Multobjectve Metaheurstcs Optmzaton n Reactve Power Compensaton ILIANA GONZÁLEZ PALAU 1, SECUNDINO MARRERO RAMÍREZ 1, ARÍSTIDES LEGRÁ LOBAINA 2. DANIEL MENDIOLA ELLIS 3 1 Electrcal Engneerng Degree, Cotopaxy Techncal Unversty, Latacunga, ECUADOR 2 Mnng Metallurgcal Insttute of Moa, Mathematcs Department, Moa, CUBA 3 Mnng Metallurgcal Insttute of Moa, Electrcal Department, Moa, CUBA lana.gonzalez@utc.edu.ec, secundno.marrero@utc.edu.ec, alegra@smm.edu.cu, dmendola@smm.edu.cu Abstract: - An effcent operaton requres delmtaton or optmzaton methods for the mprovement of techncal - economc ndcators to acheve energy effcency n ndustral networks. Ths artcle proposes a multobjectve metaheurstc optmzaton usng the metrc of Tchebycheff n the objectve functon for the reactve power compensaton, based on the modfcaton of the varable ntegraton method. An evolutonary algorthm s used to generate the ntal populaton to acheve effcent solutons. To mprove the solutons, refnng algorthms are employed usng random search condtons wth flexble stoppng crtera. The general algorthm was tested wth satsfactory results n theoretcal and real case studes. Keywords: Mult-objectve optmzaton, reactve power compensaton, evolutonary algorthms, genetc algorthms, Tchebycheff dstance, harmonc. 1 Introducton Reactve Power Compensaton s very often studed as a smple optmzaton problem wth restrctons. The soluton attends an objectve functon consttuted by a lnear combnaton of several factors, such as the nvestment and transmsson losses and constrants n forms of lmts of certan other parameters such as relablty and voltage profle. Reactve power optmzaton consttutes a sub-problem of the optmal power flow[1], whch determnes the proper adjustment of the reactve power varables lke voltage magntudes, transformers tap postons and the settng of reactve power compensaton devces to optmze the objectve functons whle satsfyng systems constrants. The reactve power optmzaton problem solutons has been focused from conventonal optmzaton technques ncludng the gradent method, quadratc programmng], nonlnear programmng, lnear programmng and nteror pont method, but the actual trends target to a mult-objectve context through evolutonary optmzaton algorthm [4,7]. In general, these technques are stll assocated wth some dffcultes, such as the handlng of problems wth characterstcs of dfferent nature, wth solutons of the reactve power optmzaton problem not lnear, dscrete. Genetc Algorthms (GA) s the most used heurstc method n the optmzaton of reactve power. The goal of GA s to look for the best soluton wthn a seres of possble solutons. By "best soluton" we refer to those that optmze a predetermned metrc for a gven problem, that s, the value closest to ths numercal value, once evaluated for a partcular evaluaton of a functon [5]. The tradtonal optmzaton algorthms usually obtan an optmal soluton, however wth the applcaton of evolutonary methods n the optmzaton, t allows the optmzaton of several smultaneous and ndependent solutons, orgnatng a set of so-called optmally effcent solutons or Pareto solutons that satsfy the Preferences of the researcher. E-ISSN: X 209 Volume 13, 2018

2 Ilana González Palau, Secundno Marrero Ramírez, Arístdes Legrá Lobana, Danel Mendola Ells Dffcultes often arse n the solutons obtaned wth the AG optmzaton method, related to the nature of the functon ftness, the form of codng solutons and the genetc dversty of the populaton. To overcome these shortcomngs, other heurstc methods have been developed that mantan some common characterstcs wth genetc algorthms, among them s the one demonstrated by Arzola [2], based on the Method of Integraton of Varables (MIV). It s a generalzaton of genetc algorthms that provde a famly of evolutonary algorthms, where members of the populaton are not necessarly updated usng a code to descrbe possble solutons, but several of them and any set of operators. The condtonal random search algorthm (CRS) [7] emerged as a modfcaton to (MIV), and was developed to solve convergence problems when appled to very large populatons. The modfcaton adopted allows: 1. The search of the populaton by subntervals, accordng to (fg 3 and 4) allowed to have a greater dversty of the ntal populaton, and ts mprovement n the whole range of solutons. 2. The search for effcent solutons n the envronment of a prevously calculated soluton s acheved n order to mprove the performance or characterstcs of the solutons. 3. Searchng for approprate populaton samples, based on the best features of the already calculated populaton, selects a subpopulaton that contans ndvduals wth favorable segments of the code whch can sgnfcantly accelerate the convergence of the method. 2. Reactve power compensaton as a mult-crtera decson makng system. The characterzaton of reactve power compensaton as a decson-makng process lead to an external analyss, whch should be realzed n order to nclude the selecton of mpled varables, s shown n Fg.1 u 1 u k coordnaton varables, d 1 d r nput data, x 1 x n decson varable and y 1 y m techncal economc ndcator. The decson varables represent the connecton optons for: x 1 (transformers wth bypass), x 2 (capactors), x 3 (flters), and x 4 (synchronous motors overexcted). The techncal - economc ndcators (y ) are selected accordng to crtera of the experts correspondng to: voltage n nodes, power factor, total harmonc dstorton, total network energy losses and the economc varable through net present value. To obtan these ndcators, approprate modellng procedures are used as models n engneerng systems. The nature of the method used allows all techncal and economc ndcators to be ncluded, or even excluded, accordng to the characterstcs of the networks. In general, the synthess of the system for decson-makng ncludes the elaboraton or selecton of the necessary tools to: 1. Selecton of the techncal-economc ndcators for gven values of the decson varables. 2. Generaton of solutons optons close to the best compromse between the selected ndcators 3. Graphc optons that allow you to know the partculartes of each soluton opton. 4. Complementary procedures of smulaton that allow to study wth more precson the partculartes of each opton generated. 3. Mathematcal formulaton of the problem. The optmzaton method uses the metrc of Tchebycheff that allows to reduce the weghted dstance of the calculated value to the desred value of each ndcator (y ), ncluded n the functon; y c y d yd m ax ω (1) E-ISSN: X 210 Volume 13, 2018

3 Ilana González Palau, Secundno Marrero Ramírez, Arístdes Legrá Lobana, Danel Mendola Ells Where: ω - weght rato, whch gves greater or less mportance to the objectves and y d, y c - desred and calculated values of dfferent objectves y. By mnmzng the equaton (1) on a set of dfferent combnatons of values of (ω), resultng n an effcent soluton space, solutons that are not worse than the others belongng to the set of all possble, for at least one of the objectves. Gven the advantages dscussed above, t s assumed as a partal objectve functon (FO) at each node = 1, m of the system, to the followng expresson: Zc, j Zd, j Z = m ax w, j (2) j Zd, j Where: Z - Value of the partal functon of node ; t accomplshed that 0 Z 1, W j - Weght coeffcent n node for each ndcator j, 0 W j 1 and W = 1 m j= 1 Zc,j, Zd,j - Calculated and desred value n node for the objectve or ndcator j. The W j coeffcent can take dfferent values dependng on the evaluated objectve. It also allows the optmzaton process to convert t from multobjectve to non-objectve when the objectves n the functon are overrdden. The desred value of the objectves depends on each partcular case, and s used as nformaton drectly assocated wth the general search, wth the restrctons n each teraton. Equaton (2) ncludes only the objectves that wll be analysed at node level: Z 1 (voltage), Z 2 (power factor) and Z 3 (voltage harmonc dstorton). Fnally for each node s: Max { λ Z } Z = (3) max Where: - weght coeffcent for node. - s the result of Z at the node for objectve () wth the worst value Once Z s obtaned, we search for the mnmum of the Z T functon (4), ncludng those ndcators that were not evaluated n the nodes, but wll be taken nto account at the level of the entre network, Z 4 correspondng to the actve losses assocated wth the compensaton and Z 5 to the economc ndcator evaluated by the prevously j calculated net present value. Z T = m ax λ 1 Z... 1,,λnZ n, λn+ 1 Z Z 4, λn + 2 (4) Where: { } - Value of the objectve functon for each confguraton of the analysed network λ λ - Weght constant for node 1 Z 5 ). n - Weght constant for ndcators (Z 4 and The weght constant for nodes, (range s between 0 and 1) corresponds to the type of node and for the ndcators that wll depend on the mportance of each partcular case. For example t s necessary to hghlght the economc ndcator, n that case t s weghted wth a weght equal to 1. Ths s possble, snce as explaned above, all weghts are regulated between [0,1]. In addton, a set of constrants must be taken nto account to ensure a feasble regon of solutons. These restrctons are gven by: 1. The capactve reactve power of the node that must be between the allowed lmts 2. Power factor of the nput node 3. Power factor of load nodes 4. Net present value (NPV) NPV NPV mn The mathematcal model, whch allows to calculate all the ndcators and ntermedate varables requred from the decson varables and the nput data, s gven by the procedures of calculaton of the load flow at the fundamental and harmonc frequency, desgned for ths research. 4. Implementaton of the method for optmzaton 4.1 Varables Integraton Method (VIM) The populaton structure s determned by the set of decson varables nvolved n the optmzaton process. To have the possble solutons, there s a codng mechansm, whch allows to assgn a sngle value to each ndvdual of the populaton, whch s representatve of ts qualty as a soluton. 5 E-ISSN: X 211 Volume 13, 2018

4 Ilana González Palau, Secundno Marrero Ramírez, Arístdes Legrá Lobana, Danel Mendola Ells It s natural that an ndvdual wth a superor atttude represents a better soluton to a problem and that, under specfc condtons, can represent a correct or even optmal soluton. From an ntal generaton of canddate solutons for the process, whch s teratve, new generatons of ndvduals are produced, each tme wth better characterstcs near the soluton of the problem. The stop crtera selecton s due to, n general case, partculartes of each concrete applcaton, but predomnatng the followngs: 1. Iteratons quantty wth no modfcatons n the populaton. 2. Dfference dstance between the best and worst soluton nferor to a preestablshed value n the populaton. 3. Predetermned number of generated solutons optons. 4. Mxed condtons. The algorthm of the VIM s showed n the Fg Random Condtonal Search Algorthm. The Random Condtonal Search (RCS) method follows the basc deas of the method (MVI) proposed by Arzola [4,5] Fg.2. Ths method s composed by a seres of algorthms wth whch the search of the ntal populaton, the mprovement of the ntal populaton and the selectve reducton of the populaton are carred out Search Algorthm of the ntal populaton The ntal populaton (IP) conssts of a set K of elements, defned by the researcher. For each teraton, the best soluton s ncluded n the populaton, and ths s done untl the populaton sze concdes wth the one ntally establshed. There are two ways of obtanng ths IP assumng that T represents the total of all possble solutons of the system: 1. Take k random ntegers between 1 and T 2. Dvde the nterval [1, T] nto K subntervals and obtan a soluton of the IP n each subnterval Intal Populaton Improvement Afterward the IP has a fxed sze, follows ts mprovements or updatng; updatng means, the comparson of the obtaned solutons n the objectve functon; select the best among them all,.e. the one that has a less value of Z; then ths new soluton s compared wth the already calculated worst soluton of the populaton and determnng a possble replacement. At the moment whch the foreseen accuracy δ s attaned, the process of generaton of random values of the populaton s restarted. For each teraton two codes of varables are generated n an nterval [A, B], obtanng three subntervals [A, x 1 ], [x 1, x 2 ] and [x 2, B]; the subnterval that contans a hgher value of Z, whch s the worst result [x 2, B], s elmnated (Fg. 2). Then, two other random values are generated wthn the subnterval obtaned [A, x 2 ]. Ths process s repeated whle the length of the nterval x 1 - x 2 remans above a gven precson (Fg. 3). Whle the populaton sze s smaller than the establshed sze, the soluton of lower Z value s added to the populaton, at each step. Once the requred populaton sze has been reached, for a new Z-value calculated, ths soluton s compared to the worst soluton and n the case of a smaller value of Z, t replaces the worst populaton soluton already found untl all the Solutons to the optmum. E-ISSN: X 212 Volume 13, 2018

5 Ilana González Palau, Secundno Marrero Ramírez, Arístdes Legrá Lobana, Danel Mendola Ells Fg. 5: Average rato of calculated Z and percent of populaton evaluated Fg. 3 Frst step for the ntal populaton search n the nterval [A, B], elmnatng the nterval [x 2, B]. Fg. 4 Second step for ntal populaton searchng n the [A, X 1 ] The algorthm leads to the search of ndvduals wth decreasng values of Z, whch means convergence of the algorthm, but t may happen that there are other solutons wth values smaller than the calculated Zx 1, to remedy ths problem, truncaton possbltes have been mplemented n the man algorthm, so that the new solutons are ncorporated as approprate. Ths algorthm has proven to be effcent when evaluated around 7% of all possble solutons (Fg. 5) Varables Codng The canddate varables to be encoded belong to the declared decson varables. The decson varables could take dfferent connectons or confguratons, defned as the (x) chromosomes. Ths nvestgaton assumed that an x chromosome s a strng character, where each character represents a transformer, a bank of capactors or a flter amongst others and every possble value of (x) depends on the defnton of the analysed element. For example an electrc system consttuted by: A transformer T, wth two taps 1 and 2 A capactor s bank C, wth two swtchable sectons 1 and 2 A flter F, wth two postons (connected or not connected). Then, the number of ndependent combnatons C p, of the confguratons, taken by these decson varables, would be; C p = 2 x 3 x 2 = 12 It s necessary to state out, the proposed codfcaton s decmal,.e., each dgt of the code accepts decmal values. Because ths code, a specal but small algorthm s requred to acheve the mplemented codfcaton n two functons provdng knowledge about x A. Wth these two functons, t could be establshed that knowng sub code (x) warrants and unvocal form of the correspondng codfcaton for each electrcal system and the recprocal statement s also true; snce the order of every element (transformer, flter and bank of capactors) of the electrcal system s nvarant durng each calculaton Stop Crtera and valdaton of optmzaton algorthm. The stop crtera used durng optmzaton s a mxed condton, n whch a number of soluton codes are calculated wthout changes n the composton of the populaton; and the dfference between the calculated Z values n the objectve functon should be less than a predetermned value (the worst and best soluton). E-ISSN: X 213 Volume 13, 2018

6 Ilana González Palau, Secundno Marrero Ramírez, Arístdes Legrá Lobana, Danel Mendola Ells The result of experments to valdate convergence of the algorthm shown n Fg. 5 were obtaned for the system of Fg. 8. The system data for smulaton and obtaned effcent solutons are shown n Tables 1 and 2 respectvely [7]. In these experments the qualty standard used to evaluate the solutons solutons corresponds to the best soluton found n the exhaustve search represented through Zexa Convergence check Fg.7. Power system to evaluate the convergence of the method. Table 1: Qc and loads nstalled n the system of Fgure 8 before optmzaton Nodes Actve Power, kw Reactve Power, kvar Capactve Power, Qc kvar Fg 6. General Algorthm for effcent solutons achevng. Each experment s performed wth an ntal populaton that has a random character, obtanng good solutons from the evaluaton of 7%, fgure 5, 7. From ths t can be deduced that for values of close to 10% of the set of all possble solutons s possble to obtan solutons that guarantee the requrements n the process of mprovements and search of effcent solutons, whch concdes wth the one proposed by Arzola Table 1 shows the actve and reactve power nstalled accordng to fg.7.the expermental smulaton results show the convergence of the method and the emergence of effcent solutons shown n the table 2. Table 2: Values of cos φ and losses for three of the effcent solutons found durng optmzaton. N 0 Calculated values Z Cos φ Qc, kvar C1 C2 C x x x x As can be seen n Table 2, the value of Z s the mnmum n confguraton 1, whch s the best soluton n terms of energy losses. E-ISSN: X 214 Volume 13, 2018

7 Ilana González Palau, Secundno Marrero Ramírez, Arístdes Legrá Lobana, Danel Mendola Ells 5. Study case Fg. 8. Electrc dagram The graphcal evoluton of the proposed algorthm (RCS) s shown to fnd the soluton of the electrc scheme n Fg. 8. Tables 3 and 4 shows the results wth three solutons 1, 2 and 3 (see fg.11), where the soluton1 has a bg value, soluton 2 has an average value and soluton 3 a small value. Table 3. Value of losses and cos φ for dfferent solutons Calculated value of (Z) n the Objectve Functon The varablty of taps n the transformer and capactors are shown n table 2. Table 4. Transformers tap poston and Qc n capactors n kvar Calculated value of (Z) n the Objectve Functon Transformer poston and value of QC n kvar T1 C1 C2 C3 Pont Pont Pont Table 5. Calculated values of four of the selected optmal solutons Soluton Calculated Values of (Z) ΔE (kwh) ΔE (kwh) Cos φ Node Node Node Soluton Soluton Soluton Cos φ n the load nodes Table 6. Change n transformer taps and capactve reactve power for optmal solutons Soluton Transformer Taps Qc n the capactors n kvar T1 C1 C2 C In Tables 5 and 6, shows four of the selected solutons as optmal, over the 10% of all the possble solutons were calculated. Ths show off possble to fndng optmal solutons (socalled Pareto solutons) from calculatng between 7% and 10 % of the total populaton, f the proposed algorthm s used and also effectveness warrantes wthout exhaustve search, all of ths, wth no doubts permts the reducton of calculaton tme. 6. Conclusons The reducton of harmonc dstorton and reactve power compensaton, can be evaluated as a problem of decson makng under multple crtera n dscrete varables, wth dfferent techncal and economc ndcators assocated, where ncludes the Tchebycheff weght dstance n the objectve functon to analyse the network to determne effcent solutons The character of the formulated task for compensaton of reactve power permt the use of procedures for the generaton of solutons based n codes evoluton, that s, any one of the proper algorthms of Varables Integraton Method. The condtonal random search algorthm generates a number of correlated solutons wth the optmzaton problem quckly and effcently, t also facltates evaluaton of multple connecton optons correctve elements n the network to facltate decson makng researcher For the best selecton of reserved ndcators can be use weghts n the objectve functon. In the case of large systems dmenson, the values must be determned through expermental tests, consderng the sgnfcance of analysed ndcator, n order to E-ISSN: X 215 Volume 13, 2018

8 Ilana González Palau, Secundno Marrero Ramírez, Arístdes Legrá Lobana, Danel Mendola Ells reduce the number of weghts coeffcents the objectve functon. It s seen n all cases obtanng effcent solutons when evaluatng a number of populaton nearly 10% of all possble solutons. The obtaned solutons could be consdered effcent respect of those obtaned wth Exhaustve Search. References 1. Prajapat, B.S. and L. Srvastava. "Mult- Objectve Reactve Power Optmzaton Usng Artfcal Bee Colony Algorthm". Internatonal Journal of Engneerng and Innovatve Technology (IJEIT), A. R. José, "Sstemas de Ingenería" Edtoral Félx Varela p M. R. Secundno, G. P. Ilana, and L. L. Arístdes. "Convergence analyss of the varables ntegraton method appled n multobjetve optmzaton n the reactve power compensaton n the electrc nets". DYNA 82 (190). Abrl, 2015, pp C. Alejandro, R. L. Demetro, and M. Juan, "Metaheurstc optmzaton appled to the calculaton of optmum load flow n electrc power grds wth mxed type control varables". Engneerng Jornal UC. Vol. 20. No. 3. Desembre pp G. P. Ilana, "Optmzaton of Reactve Power Compensaton n Industral Power Supply Networks usng a Mult Crtera Algorthm". Energy, XXVII. 6. H. G. Julo, and N. G. Rolando. "Plannng of the reactve compensaton through evolutonary programmng". IEEE Acapulco, Méxco. 7. F. D. Pres, M. A. Gomes, and C.A. Henggeler, "Mult-objectve Model for VAR Plannng n Radal Dstrbuton Networks Based on Tabu Search". IEEE Transactons On Power Systems, G. P. Ilana, and M. R. Secundno. "Reactve Power Compensaton by Evolutonary Technques". CINAREM - ISMM, Moa, Yun We L Jnwe He. "Dstrbuton system harmonc compensaton methods: An Overvew of DG- Interfacng nverters". IEEE. Industral Electroncs Magazne 8 (4). November pp S.M. Abd-Elazm, and E. S. Al, "Synergy of Partcle Swarm Optmzaton and Bacteral Foragng for TCSC Dampng Controller Desgn, J. of WSEAS Transactons on Power systems, Vol.8 No. 2, Aprl 2013, pp E. S. Al, and S. M. Abd-Elazm, Power System Stablty Enhancement va Bactera Foragng Optmzaton Algorthm, Int. Araban journal for Scence and Engneerng (AJSE), (Publsher Sprng), Vol. 38, No. 3, March 2013, pp E. S. Al, and S. M. Abd-Elazm, Optmal PSS Desgn n a Multmachne Power System va Bactera Foragng Optmzaton Algorthm, J. of WSEAS Transactons on Power systems, Vol.8 No. 4, October 2013, pp E-ISSN: X 216 Volume 13, 2018